Introduction to Groups
نویسنده
چکیده
1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ?) where G is a nonempty set and ? is a binary operation on G satisfying: (G1) a ? (b ? c) = (a ? b) ? c, ∀a, b, c ∈ G. A semigroup G is a monoid if it also satisfies: (G2) G has an element e (sometimes denoted by 1G, called the identity of G), such that e ? a = a ? e = a, ∀a ∈ G. A monoid is a group if (G3) below is satisfied. (G3) For each a ∈ G, there is an a−1 ∈ G such that a ? a−1 = a−1 ? a = e. (a−1 is called the inverse of a).
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